
where, \(W^{\ell }\)  and \(b\)  are the weights and biases, \({}^{R}x\)  is a 3D point queried at resolution \(R\)  and \(\text{mean}_{r} (\mathbf {\sigma }_D({}^{R}x))\)  is the average density at location \(^{R}x\)  from points sampled at radius \(r\) .
We hierarchically aggregate features at resolutions \(\frac{1}{2}, \frac{1}{4}\) , and \(\frac{1}{8}\)  to obtain global equivariant features. The \(\text{EQConv}^{\ell }\)  convolution layer is the same as defined in sec:background.
We employ non-linearities from [1]} that preserve equivariance and capture better equivariant features. We also compute the max-pool of point-wise features of the last layer to obtain global type-\(\ell \)  equivariant features \(F^{\ell }\) .
